Aircraft: B737
Phase of flight: Final Approach
Anomaly: ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence

NOTAMs: Indicate that ILS 28L SFO [unusable]. ATIS indicated ILS 28R approach to Visual 28L and Visual 28R. NORCAL indicated (but never cleared) we should expect TIPTOE 28L. We inquired about the LOC 28L and were told it was operational again. As a back-up to the visual 28L we initially loaded the RNAV (GPS) 28L. We modified the RNAV 28L fixes outside DUYET (FAF) by adding CHERA after SIDBY (we were on the SERFR3 STAR); removed HEMAN from the RNAV 28L and placing a discontinuity between CHERA and DUYET (FAF). The TIPTOE would have us fly heading 310 and before the discontinuity was added the course indicated 305 on the FMC. Throughout the arrival with NORCAL and before NARWL on the SERFR3 we received and complied with four separate descents; 8;000 feet; 6;000 feet and 5;000 feet and finally 3;000 feet. We queried NORCAL before EDDYY of our clearance and were cleared to fly heading 330. We believed we heard at least two aircraft cleared for the ILS 28L and queried NORCAL about the ILS 28L status--we were told it was operational again with re-opening the runway. We backed up the approach with the ILS 28L in the radios and FMC and re-briefed the most pertinent info; especially the go-around procedure if 'push came to shove' as we had heard other aircraft flying the ILS 28L. NORCAL next asked us if we had 'the bridge' and the field in sight; which we did; though truthfully there appeared to be a cloud bank on final. We were cleared visual approach runway 28L.My initial thought was it must be the cloud bank was not obscuring visibility on final or that it was south of the final approach course if NORCAL is clearing aircraft to fly visual approaches as well. Also an A319/320 was cleared for the visual 28R coming in from the east; they had us in sight and were told to maintain visual separation; we also saw the A-319/320 though never called it as he was slightly behind us and generally we fly faster but more importantly did not want to further congest the radio frequency. At about the same time as we were switched to tower; not sure which freq; NORCAL or Tower; we heard a PIREP for tops at 1;700 ft; bases at 700 ft; later it seems pretty apparent that was down final in the cloud bank; as that is pretty much what we experienced. On the tower we stated we had lost sight of the airfield and were on the ILS 28L. We were told to standby. Tower made several instructional calls to other aircraft. The last was a query to the A319/320 maintaining visual separation on us just as we entered the cloud bank. The A319/320 said he had lost us upon entering the cloud bank and tower instructed him to go around with altitude and heading instructions.Here is our quandary; we advised tower we could not see the airfield - and by extension I know the tower could not see us. We informed tower we were on the ILS 28L. I've got another aircraft in very close proximity. Tower has not given us explicit clearance to fly the ILS 28L but is aware. I elected to continue with tower's tacit understanding. I should have pressed the tower for unequivocal clearance. Period. If there was no clearance; given the proximity of other aircraft I should have pressed tower for go-around instructions. B737 Captain reported losing sight of the airport on a Visual Approach and continued on the ILS Approach without clearance.